If wave speed is dependent on medium only, then how to reconcile $v\propto f$? I have read and learnt in many places that velocity of a wave depends only on the medium through which it travels. It is clear from this that the velocity of a wave doesn't depend on the frequency of the wave because both the sound of a roaring lion and crying baby reaches our ear with the same speed. But we also know that $\text{speed} = \frac{\text{distance}}{\text{time}}\  \Rightarrow\  v=\frac{\lambda}{T}\ \Rightarrow\ v=\lambda f=\text{wavelength}\times\text{frequency}$.  In this derivation, velocity is found to be dependent on frequency.
Can anyone please explain this contradiction? Is there any fault in my perception of the concept?
 A: When pitch of a voice is changed, both the wavelength and frequency change. For example, a higher pitch will have a higher frequency (of course) but a smaller wavelength.
Okay, that being said, the independence of the propagation speed of a wave to the properties of the wave itself it an model to use in many circumstances, but not all. I don't know about acoustics, but one example that comes to my mind is in the field of optics; you might find dispersion a useful search term.
And for a more general advice, be careful when using equations involving more than two variables. Think about what is varying and what isn't; the equation alone won't tell you.
A: The equation is correct, but you incorrectly stated that wavelength "depends" on frequency.
The two (wavelength/frequency) are both part of the same property of a medium. As the wavelength increases in a given medium, the frequency decreases. If the frequency increases, the wavelength will decrease inversely.
Wavelength and frequency are inversely proportional to each other, but neigher "depends" on the other...because...they both depend on the "medium" for their values.
Hope this helps.
